Current-induced breakdown phenomena of carbon nanofibers (CNFs) for the development of carbon-based interconnects are investigated to reveal current-carrying capacity and reliability of CNF devices. Scanning transmission electron microscopy (STEM) techniques are developed to study the structural damage by current stress, including in-situ electrical measurement in STEM. The correlation between maximum current density and electrical resistivity confirms the importance of local Joule heating, showing strong coupling between electrical and thermal transport in CNFs.